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25.10: Appendix 25C Analytic Geometric Properties of Ellipses

  • Page ID
    25598
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    Consider Equation (25.3.20), and for now take \(\varepsilon<1\), so that the equation is that of an ellipse. We shall now show that we can write it as

    \[\frac{\left(x-x_{0}\right)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \nonumber \]

    where the ellipse has axes parallel to the x - and y -coordinate axes, center at (\(x_{0}\), 0) , semi-major axis a and semi-minor axis b . We begin by rewriting Equation (25.3.20) as

    \[x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}} \nonumber \]

    We next complete the square,

    \[\begin{array}{l}
    x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}}+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
    \left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
    \frac{\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}}{\left(r_{0} /\left(1-\varepsilon^{2}\right)\right)^{2}}+\frac{y^{2}}{\left(r_{0} / \sqrt{1-\varepsilon^{2}}\right)^{2}}=1
    \end{array} \nonumber \]

    The last expression in (25.C.3) is the equation of an ellipse with semi-major axis

    \[a=\frac{r_{0}}{1-\varepsilon^{2}} \nonumber \]

    semi-minor axis

    \[b=\frac{r_{0}}{\sqrt{1-\varepsilon^{2}}}=a \sqrt{1-\varepsilon^{2}} \nonumber \]

    and center at

    \[x_{0}=\frac{\varepsilon r_{0}}{\left(1-\varepsilon^{2}\right)}=\varepsilon a \nonumber \]

    as found in Equation (25.B.10).


    This page titled 25.10: Appendix 25C Analytic Geometric Properties of Ellipses is shared under a not declared license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.